[1]} specialize def:rcn-robust to the function class \({\cal F}_{\mathrm {lin}}\)  of all linear functions
\(x \mapsto v \cdot x\) 
from \(^d \rightarrow \)  (see Section 3.2 of [1]}). However, a problem with def:rcn-robust for this function class is that \( _{(,) \sim {\cal D}}\left[\phi ((v\cdot ))\right]\)  may not
have a minimum. This is not merely a technicality.
In fact, for standard loss functions
such as the logistic loss or the exponential loss,
for any linearly separable distribution \(\cal D\) ,
\( _{(,) \sim {\cal D}}\left[\phi ((v\cdot ))\right]\) 
does not have a minimum, informally, because scaling up
\(v\)  increases all of the margins, which decreases all of the
losses.Implicit bias research analyzes the effect
of the algorithm that drives \( _{(,) \sim {\cal D}}\left[\phi ((v\cdot ))\right]\)  to zero on the classification
behavior of the limiting classifier. Results of
[3]}, [4]}, [5]}
show that
different algorithms lead
to markedly different limiting classifiers. [1]} interpret thm:LS08 as saying that for \(d \ge 2\) , the pair \((\phi ,{\cal F}_{\mathrm {lin}})\)  cannot be RCN-robust for any convex potential function \(\phi \)  (see Proposition 1 of Section 3.2 of [1]}), but the fact that the minimizer typically
doesn't exist in the absence of noise interferes with this interpretation.
The unhinged loss also
cannot be minimized over \({\cal F}_{\mathrm {lin}}\) , since
by scaling up the weight vector of any linear separator, the unhinged loss can
achieve an arbitrarily large negative value.
